Quasi-steady Aerodynamic Forces Research Articles

Design, modelling, and experimental validation of a self-rotating flapping wing rotorcraft with motor–spring resonance actuation system

Compared with traditional flapping motion, the flapping wing rotor (FWR) allows rotating freedom by installing the two wings asymmetrically, which introduces rotary motion characteristics and enables the FWR to have higher lift and aerodynamic efficiency at low Reynolds number. However, most of the proposed FWRs contain linkage mechanical transmission structures, the fixed degrees of freedom of which prohibit the wings from achieving variable flapping trajectories, limiting further optimization and controller design of FWRs. In order to fundamentally address the above challenges of FWRs, this paper presents a new type of FWR with two mechanically decoupled wings, which are directly driven by two independent motor–spring resonance actuation systems. The proposed FWR has 12.4 g of system weight and 165–205 mm wingspan. In addition, a theoretical electromechanical model based on the DC motor model and quasi-steady aerodynamic forces is established, and a series of experiments are conducted in order to determine the ideal working point of the proposed FWR. It is notable that both our theoretical model and experiments exhibit uneven rotation of the FWR during flight, i.e. rotation speed dropping in the downstroke and increasing in the upstroke, which further tests the proposed theoretical model and uncovers the relationship between flapping and passive rotation in the FWR. To further validate the performance of the design, free flight tests are conducted, and the proposed FWR demonstrates stable liftoff at the designed working point.

A reduced-order multi-body model with electromechanical-aeroelastic coupling for mechanism-free ornithopters

Piezoelectric material-actuated so-called solid-state (i.e., mechanism-free) ornithopters do not need conventional motors and mechanisms, potentially saving weight and energy consumption, reducing mechanical complexity, and improving wing flexibility. In solid-state ornithopters, wing flapping can be achieved by surface-mounted piezoelectric actuators. The design of a solid-state ornithopter is a complex multi-disciplinary task, and optimization of wing topology, actuator placement, and excitation parameters requires a computationally efficient model to predict its dynamic response and performance metrics. In this paper, effectively, a novel flight simulation model is presented based on a reduced-order lumped parameter multi-body model coupled with a vortex lattice program. The proposed lumped parameter model can be represented by an equivalent multi-body mechanical model or a circuit model. The coupling considered in the lumped parameter model includes: (1) the body-wing momentum interaction, (2) bend–twist​ coupling of the piezocomposite wings, (3) fluid–structure interactions between the flapping wings, oscillating body, and the ambient fluid, and (4) electromechanical coupling between a resistive–inductive–capacitive driving circuit and the piezocomposite wings. The wing motion is described using the Rayleigh–Ritz method, in which the representative wing modes are identified as spanwise plunge, chordwise curvature, and twist. The body-wing interaction is modeled using Hamilton’s principle. The fluid-induced effects on the wings have three contributions: added mass and damping, quasi-steady aerodynamic forces obtained using the vortex lattice program, and unsteady aerodynamic forces calculated using the Theodorsen formulation, which is integrated into the wing heave and pitch degrees of freedom in the lumped parameter model. The proposed reduced-order model proposed is computationally efficient, suitable for trade studies and multi-disciplinary design optimization, and can be easily converted to state representation for controller development purposes. The capabilities of this model are demonstrated through tethered and gliding flight simulations. System properties such as circuit parameters, structural parameters, excitation frequency ratio, etc., are investigated. Phugoid and short period modes are observed in gliding flight simulations. The contribution of active flapping is found by the comparison of flight simulations with and without active flapping. A net thrust contributed by unsteady lift is demonstrated even though the thrust-lift ratio is low. The proposed model serves as an extensible framework for modeling and design optimization of ornithopters.

Effects of side ratio on energy harvesting from transverse galloping of a rectangular cylinder

Rectangular cylinders have been identified as ideal bluff bodies for galloping-based energy harvesting, while a systematic study of the effects of the side ratio (i.e., the ratio between the cylinder width and the cylinder height) remains unavailable. This study investigates the influences of the side ratio on the piezoelectric energy harvesting from the transverse galloping of a rectangular cylinder based on a representative electro-aero-mechanical model, in which the aerodynamic force is calculated by the quasi-steady theory. The existing experimental aerodynamic coefficients for rectangular cylinders with side ratios of 0.62–3.0 are utilized as inputs of the quasi-steady aerodynamic force model. The influences of the side ratio and load resistance on the onset velocity, displacement, and power output of the galloping-based energy harvester are investigated. The results show that the onset velocity of galloping is dependent on the load resistance while this dependency becomes less significant with increasing the natural frequency. The onset galloping velocity of the energy harvester decreases with increasing the side ratio, and the lowest onset velocity is achieved by a rectangular cylinder with a side ratio of around 2.50. The largest vibration amplitude is achieved by the cylinder with a side ratio of 1.62 or 2.0 at low flow velocities, while the largest vibration amplitude is always achieved by the square cylinder at high flow velocities. Therefore, the side ratio of a galloping-based energy harvester should be designed according to its working environment: the optimal side ratio is around 1.0 if the harvester is expected to work at relatively high reduced flow velocities, while the optimal side ratio is around 1.62–2.0 if the harvester is expected to be effective at relatively low reduced flow velocities. These conclusions can provide references for designing galloping-based energy harvesters with rectangular cylinders as bluff bodies.

Vibration control of cantilever blade based on trailing-edge flap by restricted control input

Vibration and control of cantilever blade with bending-twist coupling (BTC) based on trailing-edge flap (TEF) by restricted control input are investigated. The blade is a thin-walled structure using circumferentially asymmetric stiffness (CAS) configuration, with TEF embedded and hinged into the host composite structure along the entire blade span. The TEF structure is driven by quasi-steady aerodynamic forces. Vibration control is investigated based on linear matrix inequation (LMI) algorithm using restricted control input (LMI/RCI). Flutter suppression of BTC displacements and the angle of TEF (i.e. the practical control input) are illustrated, with apparently controlled effects demonstrated. The restricted control input signals are used to driven the TEF to explore the scope of the feasibility of the practical TEF angle, which is displayed by a virtual simulation platform. The platform verifies the feasibility of the hardware implementation for the control algorithms.

CFD-CSD method for coupled rotor-fuselage vibration analysis with free wake-panel coupled model

An efficient comprehensive vibration analysis method for a helicopter rotor–fuselage coupling system is presented. This loose computational fluid dynamics (CFD)/computational structural dynamics (CSD) coupling approach with a free wake–panel coupled model is used for system vibration response analysis. The CSD model of the helicopter consists of a fuselage model using a refined three dimensional (3 D) finite element model (FEM) and a rotor model consisting of nonlinear moderate deflection beam elements with 15 degrees of freedom. The unsteady Euler CFD solver is used for the flow field analysis of the entire vehicle. The induced inflow of the quasi-steady aerodynamic force is calculated with the free wake–panel coupled model, which is used to simulate rotor–fuselage aerodynamic interference. Using a full-scale helicopter as an example, the vibration responses of the typical fuselage position in hovering and level flights are analysed. When compared with the literature results and flight test data, the predictions of the proposed method are closer to the test data than those of the traditional method in hovering and low forward ratio flights, and the difference between the two methods is minimal in high forward ratio flight.

On the nonlinear effects of the mean wind force on the galloping onset in shallow cables

The critical aeroelastic behavior of horizontal, suspended, and shallow cables is analyzed via a continuous model accounting for both external and internal damping. Quasi-steady aerodynamic forces are considered, including their stationary contribution (mean wind force). This latter induces a rotation of the cable (steady swing) around the line connecting the suspension points, together with a deformation of the initial equilibrium profile under self-weight. First, by using perturbation methods, the nontrivial equilibrium configuration is analytically determined as a nonlinear function of the wind velocity. Then, the wind critical values at which bifurcations take place and the corresponding modal shapes are determined by solving a boundary value problem in the complex field. Numerical investigations are carried out to validate the perturbation solution. A preliminary nonlinear galloping analysis is also performed to verify the galloping onset in terms of non-trivial equilibrium path and critical modes. The nonlinear terms related to the fundamental path, from which bifurcations take place, play a key role revealing new insights. They are able to heavily influence the system bifurcation, making unstable configurations which instead would be aerodynamically stable without considering the effect of the mean wind force.

A Novel Aerodynamic Force and Flutter of the High-Aspect-Ratio Cantilever Plate in Subsonic Flow

This paper focuses on the derivation of the aerodynamic force for the cantilever plate in subsonic flow. For the first time, a new analytical expression of the quasi-steady aerodynamic force related to the velocity and the deformation for the high-aspect-ratio cantilever plate in subsonic flow is derived by utilizing the subsonic thin airfoil theory and Kutta-Joukowski theory. Results show that aerodynamic force distribution obtained theoretically is consistent with that calculated by ANSYS FLUENT. Based on the first-order shear deformation and von Karman nonlinear geometric relationship, nonlinear partial differential dynamical equations of the high-aspect-ratio plate subjected to the aerodynamic force are established by using Hamilton’s principle. Galerkin approach is applied to discretize the governing equations to ordinary differential equations. Numerical simulation is utilized to investigate the relation between the critical flutter velocity and some parameters of the system. Results show that when the inflow velocity reaches the critical value, limit cycle oscillation occurs. The aspect ratio, the thickness, and the air damping have significant impact on the critical flutter velocity of the thin plate.

Evaluations of Coupled Transverse-Rotational Galloping of Slender Structures with Nonlinear Effect

This paper presents evaluations of coupled two-degree-of-freedom (2DOF) galloping oscillations of slender structures with nonlinear effects, involving coupled transverse and rotational motions. The nonlinear governing equations of the coupled motions are derived by developing nonlinear formulations of the quasi-steady aerodynamic forces and damping coefficients. By solving the eigenvalue problem of a perfectly tuned 2DOF system, an analytical expression of the minimal structural damping ratio required to prevent coupled transverse-rotational galloping is established. The influences of various parameters, such as angle of wind attack, structural height, width and aspect ratio, on the onset wind velocity for the occurrence of coupled galloping are then analyzed in detail by numerical simulations. Comparisons of the onset wind velocity of the coupled 2DOF and uncoupled single-degree-of-freedom (1DOF) galloping are presented and discussed. The nonlinear vibration behavior of coupled 2DOF galloping of slender structures under varying wind speeds is also investigated.

Experiments on the transverse aeroelastic instability of a single sphere in a swing configuration under a longitudinal wind

Overhead power lines are protected from electrical discharges (that occur during storms) by the so-called “guard cables”, which are placed above the power conductors. In some places, they are provided with beacon spheres. A new instability in these cables, under a longitudinal wind, has recently been reported. The motion of the cable can damage their supports, and causing interference with the power lines, leading to cable breakage, and reducing energy availability supplied by the transmission line. The reported instabilities include oscillation and autorotation motions. In previous papers, the vortex-induced vibration has been suggested as the mechanism that can explain the system oscillations at low speed, and presented the lateral quasi-steady aerodynamic force that can justify both the autorotation motion of a sphere and the oscillations at large speed.In this paper, results of wind tunnel tests concerning a simplified system, a single sphere in a swing configuration, are reported, considering the influence of the main parameters involved: cable span and sag, beacon mass, surface roughness, and reduced wind speed.The limits of the regions corresponding to different motion behaviours (oscillation, autorotation, transitions) are presented in the wind velocity-sag map. Autorotation results have been compared with available theoretical models.

Electro-Aero-Mechanical Model of Piezoelectric Direct-Driven Flapping-Wing Actuator

We present an analytical model of a flapping-wing actuator, including its electrical, aerodynamic, and mechanical systems, for estimating the lift force from the input electrical power. The actuator is modeled as a two-degree-of-freedom kinematic system with semi-empirical quasi-steady aerodynamic forces and the electromechanical effect of piezoelectricity. We fabricated actuators of two different scales with wing lengths of 17.0 and 32.4 mm and measured their performances in terms of the stroke/pitching angle, average lift force, and average consumed power. The experimental results were in good agreement with the analytical calculation for both types of actuators; the errors in the evaluated characteristics were less than 30%. The results indicated that the analytical model well simulates the actual prototypes.

Aerodynamic modeling for large-amplitude galloping of four-bundled conductors

The galloping of overhead transmission lines, which is characterized by large-amplitude, low-frequency oscillations, occasionally occurs when the lines are subjected to ice and snow accretion. To evaluate the applicability of an aerodynamic force model based on the quasi-steady theory for simulating the galloping of overhead transmission lines, aeroelastic tests were performed in a wind tunnel using a rigid-body section model of four-bundled conductors. First, large-amplitude, low-frequency galloping was successfully simulated in a wind tunnel test using a unique support technique proposed in this paper in the vertical, horizontal, and torsional directions. Then, two different formulations of quasi-steady aerodynamic forces affecting four-bundled conductors were presented: one formulation considers all conductors as one group, whereas the other considers each subconductor independently. The two formulations are evidently different in that the aerodynamic forces attributable to torsional velocity can only be derived using the second formulation. Finally, the response amplitudes obtained from the wind tunnel tests and time-history response analyses were compared, and the results suggested that, to simulate large-amplitude, low-frequency galloping, quasi-steady aerodynamic forces should be formulated, not for all conductors together as one group, but for each subconductor independently.

Effect of engine location on flutter speed and frequency of a tapered viscoelastic wing

The flutter of a tapered viscoelastic wing carrying an engine and subjected to a follower thrust force is investigated. The wing is considered as a cantilever tapered Euler-Bernoulli beam, made of a linear viscoelastic material where Kelvin-Voigt model is assumed to represent the viscoelastic behaviour of the material. In addition, quasi-steady and unsteady aerodynamic forces models are introduced along with the follower thrust force. The mass and inertia of the engine are modelled in order to achieve more realistic behaviour of the engine upon flutter characteristics of the system. Moreover, the governing equations of motion are derived through the Extended Hamilton’s Principle. The generalized function theory is used to more accurately consider the contribution of the mass and its follower force in the governing equations. The resulting partial differential equations are solved by Galerkin method along with the classical flutter investigation approach. Parametric studies highlighting the sensitivity of the chord-wise engine location, the span-wise engine location, and the vertical engine location on the flutter speed and flutter frequency are reported. It is found that the location of the engine in the three directions play an important role in the dynamic stability of the wing.

Enhancement of flutter stability in wind turbines with a new type of passive damper of torsional rotation of blades

Classical flutter of wind turbine blades is defined as an aeroelastic instability where a torsional blade mode couples to a flapwise bending mode to result in a mutual rapid growth of the amplitude of the flapwise and torsional motions beyond certain rotational speed of the rotor. Till date, flutter instability of wind turbine rotor has not been considered to be a serious problem. However, with the advent of larger turbines fitted with relatively softer blades, classical flutter may become a more important design consideration. As an economically viable alternative to increasing the torsional rigidity (which is more expensive), the use of a type of viscous damper mounted inside the wind turbine blades is proposed to promote the flutter critical rotational speed of the rotor. The flutter critical rotational speed is determined by using a time-domain aeroelastic simulation technique based on the quasi-steady aerodynamic forces and also considering turbulence intensity of incoming wind. The optimal damper location and the optimal damping constant values are investigated for the DTU 10 MW wind turbine. It is shown that at an optimal tuning of the damper, the flutter critical rotational speed is only marginally dependent on the turbulence intensity of the incoming wind field, on the mean wind velocity and on the structural damping of the flutter torsional eigenmode. The novel tuned damper device proposed in this paper may increase the flutter critical rotational speed with more than 100%.

Experimental Investigation on Quasi-Steady and Unsteady Self-Excited Aerodynamic Forces on Cable and Rivulet

AbstractThe aerodynamic forces on the cable and rivulet are usually determined by the quasi-steady method, which cannot take signature turbulence into account. Furthermore, the oscillation of the cable and rivulet might have significant effects on the aerodynamic forces. In this study, a series of wind tunnel tests were carried out to measure the wind pressures on a cable-rivulet test model, which can keep static and moving statuses utilizing a forced vibration system developed at Hunan University. Wind pressures measured on the test model surface were then used to integrate the drag and lift forces of the cable and rivulet. The results show that vertical vibration of the test model has little effect on the pressure distribution on the cable and rivulet and the mean wind pressures are not very sensitive to the vibration of test model. On the other hand, the oscillation of the rivulet on the cable surface seems to significantly amplify the fluctuating pressures. A sudden decrease of the lift coefficient wa...

Modeling of the Large Torsional Deformation of an Extremely Flexible Rotor in Hover

This paper describes the refined modeling of the large torsional deformation of extremely flexible rotor blades with negligible structural stiffness. Equations of motion including flap bending and torsion, specifically tailored toward unconventional blades with a tip mass and experiencing large elastic twist angles, are derived using the extended Hamilton’s principle. In particular, the foreshortening of the twisted blade arising from the trapeze effect (also called bifilar effect) is explicitly included. Quasi-steady aerodynamic forces are calculated using the blade element momentum theory. The nonlinear coupled equations of motion are solved using a finite-element method. The analysis is used to predict the thrust as well as the spanwise distribution of flap bending and twist of an 18-in.-diam rotor with extremely flexible blades rotating at 1200 rpm at various collective pitch angles. These predictions are correlated with the measurement of loads obtained using a load cell, and the measurement of the deformation obtained using a noncontact optical technique called digital image correlation. It is found experimentally and analytically that tip twist angles in the range of 10 to 40 deg, depending on the blade design, are attained. This torsional deformation is dictated by the combined action of the propeller moment and the trapeze effect. A detailed explanation of the contribution of the trapeze effect to the equations of motion is presented, and it is shown that omitting the axial foreshortening due to the trapeze effect leads to a 50% error in the computation of blade-tip twist.

Galloping of a single iced conductor based on curved-beam theory

Based on the displacement interpolations and curvature–displacement relationship of spatial curved beam theory, a finite element model of iced conductor galloping is presented, which involves 3 translational degrees of freedom (DOF) and 3 rotational DOF. The nonlinearity and coupling of translational and torsional movement can be taken into account in this model. A nonlinear iteration algorithm is employed to solve for the equilibrium position under eccentric gravity loads and the quasi-steady aerodynamic forces. The linearized movement equations are derived and the stability of the equations is judged according to the initial equilibrium solution (IES). Time integration is also performed in the sub-space. An example of three thicknesses of iced C-shaped conductors shows that the update of aerodynamic forces will greatly affect the initial equilibrium position, which will result in different critical wind speeds and galloping amplitudes of an iced transmission line. Different types of galloping, such as one-loop or two-loop, will occur at different wind attack angles.

Center manifold reduction for the flutter of airfoils with gust loading

An analysis of the classic binary airfoil system perturbed by a vertical gust is presented in this paper. To simplify the derivation, the gust disturbance is replaced with white noise, and a quasi-steady aerodynamic force is used to model the aerodynamic loading. Random bifurcation in this dynamical system is studied using the ideas of center-manifold reduction, normal form and the polar coordinates transformation. Then, an explicit presentation for the stationary probability density function is found as an approximate analytical solution to the related Fokker–Planck equation. From the results, we conclude that the P-bifurcation point will not produce qualitative changes, while the D-bifurcation point will disappear under a vertical gust.

On the transient aerodynamic forces induced on heavy and light vehicles in overtaking processes

The wind sensitive vehicle design is an important issue in their development and is essentially conditioned by fuel economy and more and more by changes in environmental conditions that may be sudden (crosswind, overtaking/crossing manoeuvres). These interactions are complicated to understand because they interact on the aerodynamics, mass, suspensions, steering geometry as well as human correction. In particular, when two vehicles overtake, they mutually influence the flow-field around each other and thereby the forces and moments acting on them are affected. This is especially true when heavy and light vehicles are driven at proximity. The present study aims to clarify and to investigate experimentally the transient aerodynamic force responses in terms of quasi-steady aerodynamic forces and moments acting on heavy and light vehicles in overtaking manoeuvres and the way to reproduce these interactions in full scale automotive wind tunnels.

Application of output feedback sliding mode control to active flutter suppression of two-dimensional airfoil

The effectiveness of the sliding mode control (SMC) method for active flutter suppression (AFS) and the issues concerning control system discretization and control input constraints were studied using a typical two-dimensional airfoil. The airfoil has a trailing-edge flap for flutter control. The aeroelastic system involves a two-degrees-of-freedom motion (pitch and plunge), and the equations were constructed by utilizing quasi-steady aerodynamic forces. The control system, designed by the output feedback SMC method, was incorporated to suppress the pitch-plunge flutter. Meanwhile, the system discretization and the flap deflection constraints were implemented. Then, a classical Runge-Kutta (RK) algorithm was utilized for numerical calculations. The results indicated that the close-loop system with the SMC system could be stable at a speed above the flutter boundary. However, when the flap deflection limits are reached, the close-loop system with the simple discretized control system loses control. Furthermore, control compensation developed by theoretical analysis was proposed to make the system stable again. The parameter perturbations and the time delay effects were also discussed in this paper.

Analysis of tiltrotor whirl flutter in time and frequency domain

The whirl flutter phenomenon in a rotor is induced by in-plane hub forces, and imposes a serious limit on the forward speed. In this paper, based on Greenberg’s model, quasi-steady and unsteady aerodynamic forces are formulated to examine the whirl flutter stability for a three-bladed rotor without flexible wing modes. Numerical results are obtained in both time and frequency domains. Generalized eigenvalue solution is utilized to estimate the whirl flutter stability in the frequency domain, and Runge-Kutta method is used to analyze it in time domain. The effects of varying the pylon spring stiffness and the swashplate geometric control coupling upon the flutter boundary are investigated. An optimum pitch-flap coupling parameter is discovered through the parametric study. Aeroelastic stability boundaries are estimated with the three different aerodynamic models. It is found that the analysis with the full unsteady aerodynamics predicts the highest flutter speed.